Examples

1d Chern-Simons functionals

2d Chern-Simons functionals

Poisson σ\sigma-model

3d Chern-Simons functionals

Ordinary Chern-Simons theory

Lagrangian

Let 𝔤\mathfrak{g} be a semisimple Lie algebra. For the following computations, choose a basis{ta}\{t^a\} of 𝔤*\mathfrak{g}^* and let {ra}\{r^a\} denote the corresponding degree-shifted basis of 𝔤*[1]\mathfrak{g}^*[1].

Courant σ\sigma-model

4d Chern-Simons functionals

BF-theory and topological Yang-Mills theory

Let 𝔤=(𝔤2→∂𝔤)1\mathfrak{g} = (\mathfrak{g}_2 \stackrel{\partial}{\to} \mathfrak{g})_1 be a strict Lie 2-algebra, an L-∞ algebra concentrated in the lowesr two degrees and with the only nontrivial bracket being the binary one.

The right hand is a polynomial in the shifted generators of W(𝔤)W(\mathfrak{g}), and hence an invariant polynomial on 𝔤\mathfrak{g}. Therefore ⟨−⟩𝔤1\langle - \rangle_{\mathfrak{g}_1} is a Chern-Simons element for it.

∞\infty-Dijkgraaf-Witten theory

As we discuss below, ∞\infty-Chern-Simons theory for this setup subsumes and generalizes Dijkgraaf-Witten theory (and the Yetter model in next higher dimension). Therefore we speak of ∞\infty-Dijkgraaf-Witten theory.

Remark

The background field for ∞\infty-Dijkgraaf-Witten theory is necessarily flat.

3d Dijkgraaf-Witten theory

Chern-Simons theory for discrete structure groups. We show that this becomes a precise statement in Smooth∞GrpdSmooth \infty Grpd: the Dijkgraaf-Witten action functional is that induced from applying the ∞\infty-Chern-Simons homomorphism to a characteristic class of the form DiscBG→B3U(1)Disc B G \to \mathbf{B}^3 U(1), for Disc:∞Grpd→Smooth∞GrpdDisc : \infty Grpd \to Smooth \infty Grpd the canonical embedding of discrete ∞-groupoids into all smooth ∞-groupoids.

We also write ΓBnU(1)≃K(U(1),n)\Gamma \mathbf{B}^n U(1) \simeq K(U(1), n). Notice that this is different from BnU(1)≃ΠBU(1)B^n U(1) \simeq \Pi \mathbf{B}U(1), reflecting the fact that U(1)U(1) has non-discrete smooth structure.

Proposition

For GG a discrete group, morphisms BG→BnU(1)\mathbf{B}G \to \mathbf{B}^n U(1) correspond precisely to cocycles in the ordinary group cohomology of GG with coefficients in the discrete group underlying the circle group

Proof

By proposition the morphism is given by evaluation of the pullback of the cocycle α:BG→B3U(1)\alpha : B G \to B^3 U(1) along a given ∇:Π(Σ)→BG\nabla : \Pi(\Sigma) \to B G, on the fundamental homology class of Σ\Sigma. This is the definition of the Dijkgraaf-Witten action (for instance equation (1.2) in FreedQuinn).

and which is equipped with a vector fieldvXv_X of grade 1 whose graded Lie bracket with itself vanishes [vX,vX]=0[v_X, v_X] = 0, given, as a derivation, by thedifferential on the Chevalley-Eilenberg algebra:

The pair (X,v)(X,v) is a differential graded manifold . In this perspective the graded algebra underlying the Weil algebra of 𝔞\mathfrak{a} is the de Rham complex of XX

Ω•(X):=W(𝔞),
\Omega^\bullet(X) := \mathrm{W}(\mathfrak{a})
\,,

but the de Rham differential is just d\mathbf{d}, not the full differential dW(𝔞)=d+dCE(𝔞)d_{\mathrm{W}(\mathfrak{a})} = \mathbf{d} + d_{\mathrm{CE}(\mathfrak{a})} on the Weil algebra. The latter is thus a twisted de Rham differential on XX.

From this perspective all standard constructions of Cartan calculus usefully apply to L∞L_\infty-algebroids. Notably for vv any vector field on XX there is the contraction derivation

where {xa}\{x^a\} is a basis of generators and deg(xa)\mathrm{deg}(x^a) the degree of a generator.

We write

N:=[d,ιϵ]
N := [\mathbf{d}, \iota_\epsilon]

for the Lie derivative of this vector field. The grade of a homogeneous element α\alpha in Ω•(X)\Omega^\bullet(X) is the unique natural number n∈ℕn \in \mathbb{N} with

ℒϵα=Nα=nα.
\mathcal{L}_\epsilon \alpha = N \alpha = n \alpha
\,.

Remark.

This implies that for xix^i an element of grade nn on UU, the 1-form dxi\mathbf{d}x^i is also of grade nn. This is why we speak of grade (as in “graded manifold”) instead of degree here.

The above is indeed well-defined: on overlaps of patches the {xa}\{x^a\} of positive degree/grade transform by a degreewise linear transformation, which manifestly preserves ∑adeg(xa)xa∂∂xa\sum_a \mathrm{deg}(x^a) x^a \frac{\partial}{\partial x^a}. Notice that the ordinary (degree-0 coordinates) do not appear in this formula. And indeed the vector field locally defined by ∑axa∂∂xa\sum_a x^a \frac{\partial}{\partial x^a} (thus including the coordinates of grade 0) does not in general exist globally.

The existence of ϵ\epsilon implies the following useful statement, which is a trivial variant of what in grade 0 would be the standard Poincare lemma.

Observation

On a graded manifold every closed differential form ω\omega of positive grade nn is exact: the form

λ:=1nιϵω
\lambda := \frac{1}{n} \iota_\epsilon \omega

satisfies

dλ=ω.
\mathbf{d}\lambda = \omega
\,.

Using this differential geometric language we can now capture something very close to def. in more traditional symplectic geometry terms.

Definition

A symplectic dg-manifold of grade n∈ℕn \in \mathbb{N} is a dg-manifold (X,v)(X,v) equipped with 2-form ω∈Ω2(X)\omega \in \Omega^2(X) which is

\item non-degenerate;

closed;

as usual for symplectic forms, and in addition

of grade nn;

vv-invariant: ℒvω=0\mathcal{L}_v \omega = 0.

Example. It follows that a symplectic dg-manifold of grade 0 is the same as an ordinary symplectic manifold. In the following we are mostly interested in the case of positive grade.

Observation

The function algebra of a symplectic dg-manifold (X,ω)(X,\omega) of grade nn is naturally equipped with a Poisson bracket

where {ωab}\{\omega^{a b}\} is the inverse matrix to {ωab}\{\omega_{a b}\}.

Observation

For f∈C∞(X)f \in C^\infty(X) and v∈Γ(TX)v \in \Gamma(T X) we say that fisaHamiltonianfor is a Hamiltonian for v_ or equivalently that
_visthe[[nLab:Hamiltonianvectorfield]]of is the [[nLab:Hamiltonian vector field]] of f$ if

df=ιvω.
\mathbf{d}f = \iota_v \omega
\,.

Proposition

There is a of symplectic dg-manifolds of grade nn into symplectic Lie nn-algebroids.

Proof

The dg-manifold itself is identified with an L∞L_\infty-algebroid as in observation . For ω∈Ω2(X)\omega \in \Omega^2(X) a symplectic form, the conditions dω=0\mathbf{d} \omega = 0 and ℒvω=0\mathcal{L}_v \omega = 0 imply (d+ℒv)ω=0(\mathbf{d}+ \mathcal{L}v)\omega = 0 and hence that under the identification Ω•(X)≃W(𝔞)\Omega^\bullet(X) \simeq \mathrm{W}(\mathfrak{a}) this is an invariant polynomial on 𝔞\mathfrak{a}.

It remains to observe that the L∞L_\infty-algebroid 𝔞\mathfrak{a} is in fact a Lie nn-algebroid. This is implied by the fact that ω\omega is of grade nn and non-degenerate: the former condition implies that it has no components in elements of grade gtngt n and the latter then implies that all such elements vanish.

Proposition

Let (𝔓,ω)(\mathfrak{P},\omega) be a symplectic Lie nn-algebroid for positive nn in the image of the embedding of prop. . Then it carries the canonical L∞L_\infty-algebroid cocycle

Proposition

For (𝔓.ω)(\mathfrak{P}. \omega) a symplectic Lie nn-algebroid coming from a symplectic dg-manifold by prop. , the higher Chern-Simons action functional associated with its canonical Chern-Simons element cs\mathrm{cs} from prop. is the AKSZ Lagrangean:

Remark The AKSZ σ\sigma-model action functional interpretation of ∞\infty-Chern-Weil functionals for binary invariant polynomials on L∞L_\infty-algebroids from prop. gives rise to the following dictionary of concepts\

Proof

This is a special case of prop. , prop. in view of corollary , using that, by definition of , ω\omega is a binary and non-degenerate .

n=0n=0 – The topological particle

For XX a we may regard its cotangent bundle 𝔞=T*X\mathfrak{a} = T^* X as a Lie 0-algebroid and the canonical 2-form ω∈W(𝔞)=Ω•(X)\omega \in W(\mathfrak{a}) = \Omega^\bullet(X) as a binary invariant polynomial in degree 2.

The Chern-Simons element is the canonical 1-form α\alpha which in local coordinates is α=pidqi\alpha = p_i d q^i.

The corresponding action functional on the line

∫ℝγ*(pidqi)
\int_{\mathbb{R}} \gamma^* (p_i\, d q^i)

is the familiar term for the action functional of the particle (missing the kinetic term, which makes it “topological”).

n=1n=1 – The Poisson σ\sigma-model

Let (X,{−,−})(X, \{-,-\}) be a . Over a Darboux chart the corresponding has coordinates {xi}\{x^i\} of degree 0 and ∂i\partial_i of degree 1. We have

a Poisson-Lie algebroid valued differential form – which in components is a function ϕ:Σ→X\phi: \Sigma \to X and a 1-form η∈Ω1(Σ,ϕ*T*X)\eta \in \Omega^1(\Sigma, \phi^* T^* X) – the corresponding Chern-Simons form is

n=2n=2 – Ordinary Chern-Simons theory

We show how the ordinary arises from this perspective. So let 𝔞=𝔤\mathfrak{a} = \mathfrak{g} be a and ω:=⟨−,−⟩∈W(𝔤)\omega := \langle -,-\rangle\in \mathrm{W}(\mathfrak{g}) its Killing form invariant polynomial. For {ta}\{t^a\} a dual basis for 𝔤\mathfrak{g} we have

Often this is written in terms of the de Rham differential 2-form ddRAd_{\mathrm{dR}} A instead of the curvature 2-form FA:=ddRA+12[A∧A]F_A := d_{\mathrm{dR}} A + \frac{1}{2}[A \wedge A]. Since the former is the image under AA of dW(𝔤)d_{\mathrm{W}(\mathfrak{g})} we can alternatively write

While for this case the argument of prop. does not give a closed formula for the full equations of motions, but it still implies that field configurations FF with vanishing do solve the equations of motion. Hence that

FA=0
F_A = 0

is a sufficient condition for AA to be a point in the covariant phase space.

This is supposedly the reason for the terms “cosmo cocycle” for this condiiton: it ensures in supergravity that that flat with all fields vanishing is a (“cosmological”) solution.

References

The notion of Chern-Simons elements for L∞L_\infty-algebras and the associated imnfty\imnfty-Chern-Simons Lagrangians is due to